const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 term SNo_max_of = \x:set.\y:set.y iIn x & SNo y & !z:set.z iIn x -> SNo z -> z <= y const bij : set set (set set) prop term equip = \x:set.\y:set.?f:set set.bij x y f const omega : set term finite = \x:set.?y:set.y iIn omega & equip x y term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y const Repl : set (set set) set axiom ReplI: !x:set.!f:set set.!y:set.y iIn x -> f y iIn Repl x f const ordsucc : set set axiom ordsuccI1: !x:set.Subq x (ordsucc x) axiom ReplE_impred: !x:set.!f:set set.!y:set.y iIn Repl x f -> !P:prop.(!z:set.z iIn x -> y = f z -> P) -> P axiom bijI: !x:set.!y:set.!f:set set.(!z:set.z iIn x -> f z iIn y) -> (!z:set.z iIn x -> !w:set.w iIn x -> f z = f w -> z = w) -> (!z:set.z iIn y -> ?w:set.w iIn x & f w = z) -> bij x y f axiom equip_sym: !x:set.!y:set.equip x y -> equip y x lemma !x:set.!y:set.!f:set set.(!z:set.(!w:set.w iIn z -> SNo w) -> equip z (ordsucc x) -> ?w:set.SNo_max_of z w) -> (!z:set.z iIn y -> SNo z) -> (!z:set.z iIn ordsucc (ordsucc x) -> f z iIn y) -> (!z:set.z iIn ordsucc (ordsucc x) -> !w:set.w iIn ordsucc (ordsucc x) -> f z = f w -> z = w) -> (!z:set.z iIn y -> ?w:set.w iIn ordsucc (ordsucc x) & f w = z) -> Subq (Repl (ordsucc x) f) y -> equip (Repl (ordsucc x) f) (ordsucc x) -> ?z:set.SNo_max_of y z var x:set var y:set var f:set set hyp !z:set.(!w:set.w iIn z -> SNo w) -> equip z (ordsucc x) -> ?w:set.SNo_max_of z w hyp !z:set.z iIn y -> SNo z hyp !z:set.z iIn ordsucc (ordsucc x) -> f z iIn y hyp !z:set.z iIn ordsucc (ordsucc x) -> !w:set.w iIn ordsucc (ordsucc x) -> f z = f w -> z = w hyp !z:set.z iIn y -> ?w:set.w iIn ordsucc (ordsucc x) & f w = z claim Subq (Repl (ordsucc x) f) y -> ?z:set.SNo_max_of y z